Efficient computation of the best quadratic approximations of cubic boolean functions

Authors:
Nicholas Kolokotronis;Konstantinos Limniotis;Nicholas Kalouptsidis
Affiliations:
Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens, Greece and Department of Computer Science and Technology, University of Peloponnese, Tripo ...;Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens, Greece;Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Athens, Greece
Venue:
Cryptography and Coding'07 Proceedings of the 11th IMA international conference on Cryptography and coding
Year:
2007

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Abstract

The problem of computing best quadratic approximations of a subset of cubic functions with arbitrary number of variables is treated in this paper.We provide methods for their efficient calculation by means of best affine approximations of quadratic functions, for which formulas for their direct computation, without using Walsh-Hadamard transform, are proved. The notion of second-order nonlinearity is introduced as the minimum distance from all quadratic functions. Cubic functions, in the above subset, with maximum second-order nonlinearity are determined, leading to a new lower bound for the covering radius of the second order Reed-Muller code R(2, n). Moreover, a preliminary study of the second-order nonlinearity of known bent functions constructions is also given.